Variable Lebesgue Spaces: Foundations And Harmonic Analysis (applied And Numerical Harmonic Analysis)
by David V. Cruz-Uribe /
2013 / English / PDF
3 MB Download
This book provides an accessible introduction to the theory of
variable Lebesgue spaces. These spaces generalize the classical
Lebesgue spaces by replacing the constant exponent p with a
variable exponent p(x). They were introduced in the early 1930s but
have become the focus of renewed interest since the early 1990s
because of their connection with the calculus of variations and
partial differential equations with nonstandard growth conditions,
and for their applications to problems in physics and image
processing. The book begins with the development of the basic
function space properties. It avoids a more abstract, functional
analysis approach, instead emphasizing an hands-on approach that
makes clear the similarities and differences between the variable
and classical Lebesgue spaces. The subsequent chapters are devoted
to harmonic analysis on variable Lebesgue spaces. The theory of the
Hardy-Littlewood maximal operator is completely developed, and the
connections between variable Lebesgue spaces and the weighted
norm inequalities are introduced. The other important operators in
harmonic analysis - singular integrals, Riesz potentials, and
approximate identities - are treated using a powerful
generalization of the Rubio de Francia theory of extrapolation from
the theory of weighted norm inequalities. The final chapter applies
the results from previous chapters to prove basic results about
variable Sobolev spaces.
This book provides an accessible introduction to the theory of
variable Lebesgue spaces. These spaces generalize the classical
Lebesgue spaces by replacing the constant exponent p with a
variable exponent p(x). They were introduced in the early 1930s but
have become the focus of renewed interest since the early 1990s
because of their connection with the calculus of variations and
partial differential equations with nonstandard growth conditions,
and for their applications to problems in physics and image
processing. The book begins with the development of the basic
function space properties. It avoids a more abstract, functional
analysis approach, instead emphasizing an hands-on approach that
makes clear the similarities and differences between the variable
and classical Lebesgue spaces. The subsequent chapters are devoted
to harmonic analysis on variable Lebesgue spaces. The theory of the
Hardy-Littlewood maximal operator is completely developed, and the
connections between variable Lebesgue spaces and the weighted
norm inequalities are introduced. The other important operators in
harmonic analysis - singular integrals, Riesz potentials, and
approximate identities - are treated using a powerful
generalization of the Rubio de Francia theory of extrapolation from
the theory of weighted norm inequalities. The final chapter applies
the results from previous chapters to prove basic results about
variable Sobolev spaces.